Every identity you need — Pythagorean, double angle, half angle, sum/difference — plus a proof strategy that actually works.
| Function | Reciprocal Identity | Quotient Identity |
|---|---|---|
| sin θ | csc θ = 1/sin θ | tan θ = sin θ / cos θ |
| cos θ | sec θ = 1/cos θ | cot θ = cos θ / sin θ |
| tan θ | cot θ = 1/tan θ |
sin²θ + cos²θ = 1
From unit circle: x² + y² = 1
sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ
1 + tan²θ = sec²θ
Divide sin²θ + cos²θ = 1 by cos²θ
tan²θ = sec²θ − 1
sec²θ − tan²θ = 1
1 + cot²θ = csc²θ
Divide sin²θ + cos²θ = 1 by sin²θ
cot²θ = csc²θ − 1
csc²θ − cot²θ = 1
sin(90° − θ) = cos θ
cos(90° − θ) = sin θ
tan(90° − θ) = cot θ
cot(90° − θ) = tan θ
sec(90° − θ) = csc θ
csc(90° − θ) = sec θ
Even (cos and sec):
cos(−θ) = cos θ
sec(−θ) = sec θ
Odd (sin, tan, cot, csc):
sin(−θ) = −sin θ
tan(−θ) = −tan θ
Sine
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
Cosine
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
Tangent
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
Example: Find exact value of sin(75°)
sin(75°) = sin(45° + 30°)
= sin 45° cos 30° + cos 45° sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2) / 4
sin(2θ)
sin(2θ) = 2 sin θ cos θ
tan(2θ)
tan(2θ) = 2 tan θ / (1 − tan²θ)
cos(2θ) — three forms (memorize all)
cos(2θ) = cos²θ − sin²θ
cos(2θ) = 2cos²θ − 1
cos(2θ) = 1 − 2sin²θ
Why three forms for cos(2θ)?
Each form is useful in different situations. The second (2cos²θ − 1) is used to derive the half-angle formula for cosine. The third (1 − 2sin²θ) is used for the half-angle formula for sine. Choose the form that eliminates a variable you want to get rid of.
sin(θ/2)
sin(θ/2) = ±√[(1 − cos θ) / 2]
cos(θ/2)
cos(θ/2) = ±√[(1 + cos θ) / 2]
tan(θ/2)
tan(θ/2) = ±√[(1 − cos θ) / (1 + cos θ)] = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ
The ± sign
Determine the sign based on the quadrant of θ/2, not θ. If θ/2 is in Quadrant II, then sin(θ/2) is positive but cos(θ/2) is negative.
Example: Find exact value of cos(22.5°)
cos(22.5°) = cos(45°/2) = √[(1 + cos 45°) / 2]
= √[(1 + √2/2) / 2] = √[(2 + √2) / 4]
= √(2 + √2) / 2
(Positive because 22.5° is in Quadrant I)
Golden Rule
Work on ONE side only. Never move terms across the equal sign. Transform one side until it matches the other. Treat it as a one-way simplification, not an equation you can solve.
Replace tan, cot, sec, csc using reciprocal/quotient identities. This puts everything in terms of two functions.
Find common denominators, split fractions, or combine. Look for terms that cancel.
When you see sin²+cos², 1−sin², 1+tan², etc. — substitute immediately with the Pythagorean identity.
Factor sin²θ−cos²θ as (sinθ+cosθ)(sinθ−cosθ). Factor out common terms.
If you see (1±sinθ) or (1±cosθ) in a denominator, multiply numerator and denominator by the conjugate.
Worked Example: Verify sin θ / (1 − cos θ) = (1 + cos θ) / sin θ
Work on left side. Multiply by conjugate (1 + cos θ)/(1 + cos θ):
= sin θ · (1 + cos θ) / [(1 − cos θ)(1 + cos θ)]
= sin θ · (1 + cos θ) / (1 − cos²θ)
Substitute 1 − cos²θ = sin²θ:
= sin θ · (1 + cos θ) / sin²θ
= (1 + cos θ) / sin θ ✓
The three Pythagorean identities are: (1) sin²θ + cos²θ = 1 (the fundamental one); (2) 1 + tan²θ = sec²θ (divide the first by cos²θ); (3) 1 + cot²θ = csc²θ (divide the first by sin²θ). All three come from the unit circle equation x² + y² = 1.
Double angle formulas: sin(2θ) = 2·sin θ·cos θ. Cosine has three equivalent forms: cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. The last two forms come from substituting the Pythagorean identity. tan(2θ) = 2·tan θ / (1 − tan²θ).
Work on one side only (usually the more complex side). Common strategies: (1) Convert everything to sin and cos; (2) Factor common expressions; (3) Multiply by a conjugate (e.g., multiply (1−sin θ)/(1−sin θ)); (4) Use Pythagorean identities to substitute; (5) Separate a fraction into individual terms. Never move terms across the equal sign — each side must be simplified independently.
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